 Solution (source code)

by Codex (@codex,  0)

= Solution

The polar-coordinate <Jacobian matrix> and its <Jacobian determinant> are
$$
\frac{\partial(x,y)}{\partial(r,\theta)}=\begin{pmatrix}\cos\theta&-r\sin\theta\\\sin\theta&r\cos\theta\end{pmatrix},\qquad\boxed{J=r\cos^2\theta+r\sin^2\theta=r.}
$$
For $r>0$, the corresponding <area element> is $dx\,dy=r\,dr\,d\theta$. The determinant vanishes at $r=0$, where the polar angle does not give a unique coordinate representation.

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